Dynamical Analysis Of Two Kinds Of Stochastic Infectious Disease Models

The development of humanity cannot be separated from the fight against infectious disease,which has always been the public enemy of all mankind.Dating back to the second century A.D,the outbreak of the Antonine plague in the Roman Empire,and the recent global epidemic of"Novel Coronavirus Pneumonia" both indicate that it is a difficult task to prevent and control the outbreak of infectious diseases.In this thesis,two kinds of stochastic infectious disease models are established by considering the infectious characteristics of diseases.And the dynamic properties of the models are analyzed by using the theory and method of stochastic differential equation,including the existence and uniqueness of global solutions,the extinction and persistence of diseases and the existence of stationary distribution.There are four chapters in this thesis.In Chapter 1,we introduce the research background and current situation,and give the relevant basic knowledge.In Chapter 2,considering the impact of stochastic environment noise on infection rate,a stochastic SIS epidemic model with nonlinear incidence rate is proposed and analyzed.Firstly,for the corresponding deterministic system,the threshold which determines the extinction or permanence of the disease is obtained by analyzing the stability of the equilibria.Then,for the stochastic system,the global dynamics is investigated by using the theory of stochastic differential equations;especially the threshold dynamics is explored when the stochastic environment noise is small.The results show that the condition for the epidemic disease to go to extinction in the stochastic system is weaker than that of the deterministic system,which implies that stochastic noise has a significant impact on the spread of infectious diseases and the larger stochastic noise is conducive to controlling the epidemic diseases.To illustrate this phenomenon,we give some computer simulations with different intensities of the stochastic noise.In Chapter 3,a stochastic SIRC epidemic model for Influenza A is proposed and investigated.First,we show that there is a unique global positive solution of the system.Second,the extinction of the disease is explored and the sufficient conditions for extinction of the disease are obtained.And then the existence of a unique ergodic stationary distribution of the positive solutions to the system under certain sufficient conditions is discussed by constructing a suitable stochastic Lyapunov function.Finally,numerical simulations are employed to illustrate the theoretical results.In Chapter 4,we summarize the full text and give some questions worthy of further study.